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How Projections Work

What lies behind a projection? Which rules tell the
cartographer where coastlines are plotted? How can one define
a mapping (mathematically, a conversion between
two sets of values) from sphere coordinates to planar points?
How good is that transformation?

"Projective"/Perspective/Geometric vs. "Algorithmic" Projections

Some projections of the azimuthal, cylindrical
and conic
families have a direct geometric interpretation as light rays
projected from a source intercept the Earth and,
according to laws of perspective, "draw" its
features on a surface. The latter may be a plane, yielding the map
itself, or an intermediate shape like a
cylindrical or conical shell.

On the other
hand, many projections are only distantly inspired by geometric
principles. For instance,
Mercator's
cylindrical projection can't be visualized as a
perspective process unless:

light rays don't follow straight trajectories, or

the light source is not a point or straight line, or

the projection surface is not a simple tube

In all three cases the complexity negates the usefulness of a
perspective model.

Indeed, many projections have simply no geometric or physical
interpretation, and are described purely by mathematical
formulae. I.e., the cartographer devises a
spherical-to-flat mapping according to some desirable but arbitrary
property or constraint.

Forward and Inverse Formulas

Perspective or not, a projection can be defined by two sets
of mapping equations:

a forward or direct relation converts
polar coordinates
(longitude ,
latitude , Earth radius or its equivalent for the ellipsoidal case) to
Cartesian coordinates (abscissa or horizontal distance
from the origin, ordinate or vertical distance )

an inverse relation performs the opposite
transformation

Coordinate transformations defined
by mapping relations

Usually those relations are not functions, as the same
point on the sphere may be represented by several points on the map.
Instead of Cartesian distances, plane polar coordinates (radius
,
angle ) can be used, being in fact easier to express for many
projections.

Although not generally presented here, inverse mapping makes possible
calculating the geographic location given a point on a map or an aerial/satellite
photograph. Thus, it is relevant to several problems, like interactive
mapping applications which, when the user clicks on the map, respond
depending on which building, street, city or whatever is
georeferenced there. It is of course important for
reprojecting, i.e., converting an
already projected map to other projections, and translating between
different geographic databases.

Deriving Projection Formulae

Even those without an interest in mathematics could get a fresh
insight on the geographical sciences by understanding a
projection formula or two; however, the reader can instead
skip
ahead to the main projection groups.

The next sections sketch the actual process for deriving
mapping formulae for a few projections:

the azimuthal
orthographic projection, purely
geometric, can be understood by anyone; basic trigonometry
is involved only for actual computation

The shape of the map is defined beforehand for all but the
Winkel, Kavrayskiy VII and (depending on the approach) Sanson-Flamsteed projections, where it is a
consequence of the projections's constraints. Only Mollweide's,
Craster's, azimuthal equal-area and Winkel derivations require
basic calculus, numerical methods, or both.

Some Conventions

Instead of commonplace degrees, minutes and seconds of arc, in
cartographic mathematics any angles including latitude and longitude are
more usefully measured in radians, since the length of
a circular arc — thus distances along a great circle —
can be directly calculated by its radius
multiplied by the angle in radians. E.g., a straight angle of
180° is equivalent to radians;
all points at latitudes 60°N and 60°S are
units away from the Equator.
Northern latitudes and eastern longitudes are arbitrarily
considered positive angles; e.g., 45°S is expressed as .

Both forward and inverse mapping require a
scaling factor,
which determines but must not to be confused with the
map's scale or scale factor:
while the scaling factor is a single constant, the scale and
scale factor unavoidably change depending on location and direction
across the map.
Equations included here express the scaling factor with the
constant ,
which is a small non-negative fraction of the Earth's actual radius.